A Global Galerkin Method for Solving the Exterior Neumann Problem for the Helmholtz Equation Using Panich's Integral Equation Approach

Abstract

We describe a boundary integral equation that solves the exterior Neumann problem for the Helmholtz equation in three dimensions for smooth obstacles that can be described globally in spherical coordinates. The unique solution is found by approximating the Fredholm integral equation of the second kind with the Galerkin method, where the basis functions are spherical harmonics. This leads to a fast method for small and medium wave numbers, since the system of equations is of size smaller than 64. To smooth the integrand a new map is introduced which significantly improves the accuracy. Numerical results for several smooth surfaces are presented.